| L(s) = 1 | + 15.2·2-s − 27·3-s + 104.·4-s + 244.·5-s − 411.·6-s − 1.05e3·7-s − 358.·8-s + 729·9-s + 3.73e3·10-s − 3.06e3·11-s − 2.82e3·12-s + 1.45e4·13-s − 1.60e4·14-s − 6.60e3·15-s − 1.88e4·16-s + 3.66e4·17-s + 1.11e4·18-s − 3.21e4·19-s + 2.55e4·20-s + 2.83e4·21-s − 4.68e4·22-s − 7.83e4·23-s + 9.68e3·24-s − 1.82e4·25-s + 2.21e5·26-s − 1.96e4·27-s − 1.09e5·28-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 0.577·3-s + 0.816·4-s + 0.875·5-s − 0.778·6-s − 1.15·7-s − 0.247·8-s + 0.333·9-s + 1.18·10-s − 0.695·11-s − 0.471·12-s + 1.83·13-s − 1.56·14-s − 0.505·15-s − 1.14·16-s + 1.81·17-s + 0.449·18-s − 1.07·19-s + 0.714·20-s + 0.668·21-s − 0.937·22-s − 1.34·23-s + 0.142·24-s − 0.232·25-s + 2.46·26-s − 0.192·27-s − 0.945·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.645588293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.645588293\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
| good | 2 | \( 1 - 15.2T + 128T^{2} \) |
| 5 | \( 1 - 244.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.05e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.06e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.45e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.66e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.21e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.83e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.80e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.26e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.57e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.65e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.16e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.41e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.42e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.29e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.04e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 5.60e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.75e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.79e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.14e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.89e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.32e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.52e7T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08084828650847020596743353215, −9.176071764024812400286276609910, −7.88168873020656913107017030342, −6.27127737009724636467539409302, −6.13017340856705384986189473544, −5.41875232684296282739571832662, −4.07112354869979428545819541488, −3.34943815044578572817674126700, −2.17362773462488591491002237100, −0.70001953838604229896628547731,
0.70001953838604229896628547731, 2.17362773462488591491002237100, 3.34943815044578572817674126700, 4.07112354869979428545819541488, 5.41875232684296282739571832662, 6.13017340856705384986189473544, 6.27127737009724636467539409302, 7.88168873020656913107017030342, 9.176071764024812400286276609910, 10.08084828650847020596743353215
